From the given table, let's find an exponential function to model the data.
To write the exponential function, apply the formula:
[tex]f(x)=ab^x[/tex]Where:
b is the rate of change.
We have:
[tex]f(x)=7=ab^1[/tex]Now substitute (7, 522) for the values of x and f(x):
[tex]522=ab^7[/tex]Divide both equations to find b:
[tex]\begin{gathered} \frac{ab^7}{ab^1}=\frac{522}{7} \\ \\ b^6=74.57 \\ \\ b=2.04 \end{gathered}[/tex]The value of b is 2.04.
To find the value of a, we have:
[tex]f(x)=a(2.04)^x[/tex]Substituet (7, 522) for values of x and f(x):
[tex]\begin{gathered} 522=a(2.04)^7 \\ \\ 522=147.032a \end{gathered}[/tex]Divide both sides by 147.032a:
[tex]\begin{gathered} \frac{522}{147.032}=\frac{147.032a}{147.032} \\ \\ 3.56=a \\ \\ a=3.56 \end{gathered}[/tex]Therefore, the exponential function to model the data is:
[tex]f(x)=3.56(2.04)^x[/tex]ANSWER:
[tex]f(x)=3.56(2.04)^x[/tex]